Intro
Issue
In order to simulate sylviculture with TROLL we need to implement a new sylviculture module inside TROLL model code. A first litterature review was completed by an interview with Laurent Descroix of the Office Nationale des Forêts. We discovered that rotten trees were not random and seemed to depend both on tree species and diameter. This document presents modelling of relation between rotten trees and their species and diameter.
In fact we have two different questions:
- Predict if a tree will be probed as rotten (models M, see rotten 2)
- Predict how much of tree volume is rotten (models N, see rotten 3)
| N | Model |
|---|---|
| \(N\) | \(V_f \sim \mathcal{logN} (log[(\beta*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
| \(N_{p}\) | \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p)*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
| \(N_{s}\) | \(V_f \sim \mathcal{logN} (log[((\beta + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
| \(N_{p,s}\) | \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\) |
We tested models N detailed in following tabs to find the better trade-off between:
- Complexity (and number of parameters)
- Convergence
- Likelihood
Results are shown for each models in each model tabs and summarized in Conclusion tab.
Graph
Trees estimated dbh and final softwood product.
\(N\)
Model
\(N\): \(V_f \sim \mathcal{logN} (log[(\beta*dbh^2)*(1 - Pr*\rho)], \sigma)\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (15.383*dbh^2)*[1-Pr*0.189]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(N_{p}\)
Model
\(N_{p}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p)*dbh^2)*(1 - Pr*\rho)], \sigma)\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (15.259*dbh^2)*[1-Pr*0.228]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(N_{s}\)
Model
\(N_{s}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (14.731*dbh^2)*[1-Pr*0.152]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(N_{p,s}\)
Model
\(N_{p,s}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p + \beta_s)*dbh^2)*(1 - Pr*\rho)], \sigma)\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (14.782*dbh^2 + 1)*[1-Pr*0.209]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
Conclusion 1
Table 3: Models summary.Adding random effect for plot and/or species (\(\beta_p\), \(\beta_s\)) increased likelihood and decreased model variance (\(\sigma\)). Consequently best model is \(N_{p,s}\). Stille we can try to improve this model by adding random effect for species and/or plot to the hyperparameter \(\rho\).
| L | Model |
|---|---|
| \(L\) | \(\rho = \theta *dbh^2\) |
| \(L_{p}\) | \(\rho = (\theta + \theta_p) *dbh^2\) |
| \(L_{s}\) | \(\rho = (\theta + \theta_s) *dbh^2\) |
| \(L_{p,s}\) | \(\rho = (\theta + \theta_p + \theta_s) *dbh^2\) |
\(L\)
Model
\(L\): \(\rho = \theta *dbh^2\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (13.715*dbh^2)*[1-Pr*(0.506*dbh^2)]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(L_{p}\)
Model
\(L_{p}\): \(\rho = (\theta + \theta_p) *dbh^2\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (14.308*dbh^2)*[1-Pr*(0.502*dbh^2)]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(L_{s}\)
Model
\(L_{s}\): \(\rho = (\theta + \theta_s) *dbh^2\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (13.56*dbh^2)*[1-Pr*(0.532*dbh^2)]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(L_{p,s}\)
Model
\(L_{p,s}\): \(\rho = (\theta + \theta_p + \theta_s) *dbh^2\)
Model Predictions.
Parameters markov chains. Light grey area represents warmup iterations.
Convergence
\(P_{rotten} = (13.849*dbh^2)*[1-Pr*(0.561*dbh^2)]\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
Conclusion 2
Adding random effect for plot and/or species (\(\theta_p\), \(\theta_s\)) increased likelihood and decreased model variance (\(\sigma\)). Consequently best model is \(N_{p,s}\) associated to \(L_{p,s}\): \(V_f \sim \mathcal{logN} (log[((\beta + \beta_p + \beta_s)*dbh^2)*(1 - Pr*((\theta + \theta_p + \theta_s) *dbh^2))], \sigma)\)